Shape classification based on interpoint distance distributions

According to Kendall (1989), in shape theory, The idea is to filter out effects resulting from translations, changes of scale and rotations and to declare that shape is “what is left”. While this statement applies in principle to classical shape theory based on landmarks, the basic idea remains also when other approaches are used. For example, we might consider, for every shape, a suitable associated function which, to a large extent, could be used to characterize the shape. This finally leads to identify the shapes with the elements of a quotient space of sets in such a way that all the sets in the same equivalence class share the same identifying function. In this paper, we explore the use of the interpoint distance distribution (i.e. the distribution of the distance between two independent uniform points) for this purpose. This idea has been previously proposed by other authors [e.g., Osada et al. (2002), Bonetti and Pagano (2005)]. We aim at providing some additional mathematical support for the use of interpoint distances in this context. In particular, we show the Lipschitz continuity of the transformation taking every shape to its corresponding interpoint distance distribution. Also, we obtain a partial identifiability result showing that, under some geometrical restrictions, shapes with different planar area must have different interpoint distance distributions. Finally, we address practical aspects including a real data example on shape classification in marine biology.